Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,5,7)(17,18,22,24)(19,20,21,23);(1,3,6,5)(2,7,4,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,21)(18,23,24,20);(3,5)(7,8)(10,14)(12,15)(18,24)(19,21);(17,19,22,21)(18,20,24,23);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,23)(18,19,24,21);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,11,14,16)(17,21,22,19)(18,23,24,20)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233566, #72233585, #72233609, … |
| maximal normal subgroup of | #72233566, #72233585, #72233609, … |
| maximal quotient of | #72262156, #72297739, #72297896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7653914);